Improper integrals for stability theory of multidimensional phase systems

Abstract

A class of ordinary differential equations describing the dynamics of multidimensional phase systems
with a countable equilibrium position with periodic nonlinear functions from a given set is
considered. Such uncertainty of right side of the differential equation gives rise to the nonuniqueness
of the solution, which leads to the study of the properties of solutions of equations with
differential inclusions. A completely new approach to the study of the properties of the solution of
dynamical systems with a countable equilibrium position with incomplete information on nonlinearities
is proposed. By a nonsingular transformation, the original system is reduced to a special
kind, consisting of two parts. The first part of the differential equations is solvable with respect to
the components of the periodic function and the second part does not contain nonlinear functions.
The properties of solutions are studied, estimates for the solutions of the original system and the
transformed system are obtained and their boundedness is proved. Identities with respect to the
components of the nonlinear function are obtained and their relation to the phase variables is
established. The properties of quadratic forms with respect to phase variables and derivatives are
studied. The estimates of improper integrals along the solution of the system are obtained for two
cases: when the values of the integrals of the components of the nonlinear function in the period
are zero; When the values of the integrals in the period are different from zero. These results can
be used to obtain conditions for global asymptotic stability of multidimensional phase systems.